Recall that W. Ackermann verified what in modern terms we call the bi-interpretability of and , where the latter is (first-order) Peano arithmetic, and the former is finite set theory, the result of replacing in the axiom of infinity with its negation (and with foundation formulated as the schema of -induction). The reference is

I have written about this before. Briefly, one exhibits (definable) translations between the collection of hereditarily finite sets and and verifies that the translation extends to a definable translation of the relations, functions and constants of the language of each structure in a way that verifies that holds in the translation of and verifies that holds in the translation of . Recall that consists of those sets whose transitive closure is finite, that is, is finite, and all its elements are finite, and all the elements of its elements are finite, and so on. Using foundation, one easily verifies that , that is, it is the collection of sets resulting from iterating the power-set operation (any finite number of times) starting from the empty set.

In the direction relevant here, one defines a map by

One easily verifies, using induction on the set-theoretic rank of the sets involved, that this recursive definition makes sense and is injective (and, indeed, bijective).

Of course this argument uses foundation. In the D’Agostino-Policriti-Omodeo-Tomescu paper they consider instead the theory resulting from replacing foundation with the anti-foundation axiom, and proceed to describe a suitable replacement for that injects (codes) into the real numbers. They do quite a bit more in the paper but, for the coding itself, I highly recommend the nice review by Randall Holmes in MathSciNet, linked to above.

The anti-foundation axiom became known thanks to the work of Peter Aczel, and it is his formulation that I recall below, although it was originally introduced in work of Forti and Honsell from 1983, where they call it . Aczel’s presentation appears in the excellent book

Given a binary relation , its field is the union of its domain and codomain. A decoration of is a function satisfying

for all . When is and the sets in question are well-founded, the only decoration is the identity. Similarly, any well-founded relation admits a unique decoration. Define as the statement that any binary (whether well-founded or not) admits a unique decoration.

In with foundation replaced with one can prove the existence of many non-well-founded sets. One of the appealing aspects of is that the resulting univere is actually quite structured: Other anti-foundation axioms allow the existence of infinitely many Quine atoms, sets such that , for instance. Under , there is exactly one such , usually called . The axiom is sometimes described as saying that it provides solutions to many “equations” among sets. For instance, consider the system of equations and . Under the system has as its unique solution. Note that assuming , is in , as are many other non-well-founded sets.

Here is the open question from the D’Agostino-Policriti-Omodeo-Tomescu paper: Work in set theory with instead of foundation. Is there a unique, injective, function satisfying

for all ?

Note that there is a unique such on the well-founded hereditarily finite sets, and it is in fact injective. In general, existence, uniqueness and injectivity of appear to be open. The claim that there is such a function is a statement about solutions of certain equations on the reals, and the claim that is unique requires moreover uniqueness of such solutions. The expectation is that is transcendental for all non-well-founded hereditarily finite but, even assuming this, the injectivity of seems to require additional work.

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The only reference I know for precisely these matters is the handbook chapter MR2768702. Koellner, Peter; Woodin, W. Hugh. Large cardinals from determinacy. In Handbook of set theory. Vols. 1, 2, 3, 1951–2119, Springer, Dordrecht, 2010. (Particularly, section 7.) For closely related topics, see also the work of Yong Cheng (and of Cheng and Schindler) on Harr […]

As other answers point out, yes, one needs choice. The popular/natural examples of models of ZF+DC where all sets of reals are measurable are models of determinacy, and Solovay's model. They are related in deep ways, actually, through large cardinals. (Under enough large cardinals, $L({\mathbb R})$ of $V$ is a model of determinacy and (something stronge […]

Throughout the question, we only consider primes of the form $3k+1$. A reference for cubic reciprocity is Ireland & Rosen's A Classical Introduction to Modern Number Theory. How can I count the relative density of those $p$ (of the form $3k+1$) such that the equation $2=3x^3$ has no solutions modulo $p$? Really, even pointers on how to say anything […]

(1) Patrick Dehornoy gave a nice talk at the Séminaire Bourbaki explaining Hugh Woodin's approach. It omits many technical details, so you may want to look at it before looking again at the Notices papers. I think looking at those slides and then at the Notices articles gives a reasonable picture of what the approach is and what kind of problems remain […]

It is not possible to provide an explicit expression for a non-linear solution. The reason is that (it is a folklore result that) an additive $f:{\mathbb R}\to{\mathbb R}$ is linear iff it is measurable. (This result can be found in a variety of places, it is a standard exercise in measure theory books. As of this writing, there is a short proof here (Intern […]

The usual definition of a series of nonnegative terms is as the supremum of the sums over finite subsets of the index set, $$\sum_{i\in I} x_i=\sup\biggl\{\sum_{j\in J}x_j:J\subseteq I\mbox{ is finite}\biggr\}.$$ (Note this definition does not quite work in general for series of positive and negative terms.) The point then is that is $a< x

The result was proved by Kenneth J. Falconer. The reference is MR0629593 (82m:05031). Falconer, K. J. The realization of distances in measurable subsets covering $R^n$. J. Combin. Theory Ser. A 31 (1981), no. 2, 184–189. The argument is relatively simple, you need a decent understanding of the Lebesgue density theorem, and some basic properties of Lebesgue m […]

Given a class $S$, to say that it can be proper means that it is consistent (with the axioms under consideration) that $S$ is a proper class, that is, there is a model $M$ of these axioms such that the interpretation $S^M$ of $S$ in $M$ is a proper class in the sense of $M$. It does not mean that $S$ is always a proper class. In fact, it could also be consis […]

As the other answers point out, the question is imprecise because of its use of the undefined notion of "the standard model" of set theory. Indeed, if I were to encounter this phrase, I would think of two possible interpretations: The author actually meant "the minimal standard model of set theory", that is, $L_\Omega$ where $\Omega$ is e […]